Permanence Of A Lotka-Volterra System And Oscillation Criteria For Delay Difference Equations With Impulsive Effect

In population ecology, the Lotka-Volterra model is a fundamental one. It can be classified three types according to ecological meaning:predator-prey, competition, cooperation. Especially, predator-prey model has always been a hotspot of research. In this chapter, we investigate a Lotka-Voltrra predator-prey system with impulsive effect on the predator. By using comparison methods and analyzing right functionmethod, we prove extinction and permanence of the system. Furthermore,by applying Lakmeche and Arino′s researching results: the impulsive bifurcation theory, we show that the existence of a positive periodicsolution. In this chapter, we will develop the Lotka-Volterra predator-prey model with Holling type Ⅳfunctional responses by introducing a constant periodic impulsive immigration for the predator. That is 21 1 1 1 1 2112 12 2 2 2111 20 01 02, ,,( ) 0, ( ) , ,(0 ) ( , )T ,x x r a x xi cxx at nx x r xi a xx ax t x t b t nx x x xττ+?? ?? ??? = ? ? ? ?≠?????????? ? = =????????? ? ?+ += ++ ??????+???=?? = =&& (1) where x1 (t ), x2 (t ) are the densities of the prey and predator at time trespectively, ? xi ( t)= xi (t + )? xi (t ),i = 1,2, r1 is the intrinsic growth rate of prey, r2 is the death rate of predator, a1 is the rate of intraspecific competition of prey, c , a2 , i , a , b, x0 1 ,x0 2 are positive, τis theperiod of the impulsive immigration effect. The main conclusion are given by Theorem 3.1 There exists a constant M > 0 such that xi (t ) ≤M, i = 1,2, for each solution x (t ) = ( x1 (t ), x2 (t )) of (1) with all t large enough. Theorem 3.2 Assume x (t ) be any solution of (1). Then x1 (t ) →0,x2 (t ) →x2 ? (t ) as t →+∞provided one of the following conditions is satisfied. (h1): 21 2 ( ),brr MiM acτ+ +> (h2): 2br1 r2 ( MiM a)cτ+ += and 022,1 exp( )xb≥? ? rτwhere M is defined in Theorem 3.1, x2 ? (t )= 22exp( ( )) ,1 exp( )b r t nrττ? ?? ? t∈( nτ, , ( n + 1)τ], n ∈N, 22(0 )1 exp( )xbrτ? + = ? ? .Theorem 3.3 If the following condition holds (h3): 21 21 2( )min ,brr MiM a rrc c< ???? τ+ +τ??????? ??? Then system (1) is permanent. Theorem 4.2 system (1) has a stable positive periodic solution, if condition (h4): aa1 ≥r1 holds, 01 2bcτ> τ= arr and is closing to τ0. Chapter two Oscillation criteria for impulsive difference equations We consider the impulsive delay difference equation 11( ( ( 1)) ) ( , ( )) 0, 0, , ,( ( )) ( ( ( 1)) ),k kn kn k k n ka x n f n x n l n n k Na x n I a x nσασσαα?α?????? ? ? ? ?= + ? ?? = > ≠∈(1) in which ?x ( n ) = x ( n + 1) ? x ( n),?αx ( n ) = x ( n + 1) ? αx ( n),σis the quotient of any two positive odd numbers, l ∈N, N is the set of natural number,0 ≤n0≤n1 ≤L ≤nk ≤L , and lk i→m∞nk= +∞. Throughout this chapter, assume that the following conditions hold (ⅰ) uf ( n, u ) > 0(u ≠0) and there exists a nonnegative sequence { pn } and a function u , such that f (u nσ, u ) ≥pn; (ⅱ) there exist positive numbers bk , bk , such that bk ≤I kx( x) ≤bk, k ∈N; (ⅲ) 0{a n }∞n is a positive sequence. The main conclusions are given by Theorem 1 Assume the following conditions hold (h1): (ⅰ)-(ⅲ) hold; (h2): for all sufficiently large n j( ≥n1) and n →∞, such that110jjj k jjn n mn nkn n n mmn mbaσσ?α? ?≤≤+=+→+∞∏∑, and if 1,( )j k j knii n i n n n i kp nbασ= + ≠< ≤∑∏→+∞→∞holds for all sufficiently large n j. Then every solution of (1) is oscillatory. Corollary 1 Assume that (h1),(h2) hold and there exists a positive integer k 0 such that 1α≥(b k )σfor k ≥k0. If k,nn n k N+∞p≠∈∑= +∞. Then every solution of (1) is oscillatory. Corollary 2 Assume that (h1),(h2) hold and there exists a positive integer k 0, for k ≥k0, such that 1k 1 ( k)kn bnα ≥+σand k,nn n k N+∞nσp≠∈∑= +∞. Then every solution of (1) is oscillatory.